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Parametric Equations and Polar Coordinates

  • Understand the concept of parametric equations and their applications.
  • Convert between parametric and Cartesian equations.
  • Learn the fundamentals of polar coordinates.
  • Convert between polar and Cartesian coordinates.
  • Perform calculus operations on parametric and polar equations.

Definition of Parametric Equations

A curve in the plane can be represented by a set of **parametric equations**:

\[ x = f(t), \quad y = g(t), \quad t \text{ is the parameter} \]

For example, the parametric equations:

\[ x = \cos t, \quad y = \sin t, \quad 0 \leq t \leq 2\pi \]

represent a unit circle.

Eliminating the Parameter

To convert from parametric to Cartesian form, solve for \( t \) in terms of \( x \) or \( y \).

Example: Given \( x = 3t \) and \( y = 2t + 1 \), eliminate \( t \):

\[ t = \frac{x}{3} \] \[ y = 2 \left(\frac{x}{3}\right) + 1 = \frac{2x}{3} + 1 \]

Introduction to Polar Coordinates

In **polar coordinates**, a point is represented as \( (r, \theta) \), where:

  • \( r \) is the distance from the origin.
  • \( \theta \) is the angle from the positive x-axis.

For example, \( (2, \frac{\pi}{4}) \) means the point is 2 units away at an angle of \( 45^\circ \).

Conversion Between Polar and Cartesian Coordinates

To convert:

  • From Cartesian to Polar: \[ r = \sqrt{x^2 + y^2}, \quad \theta = \tan^{-1} \left(\frac{y}{x}\right) \]
  • From Polar to Cartesian: \[ x = r\cos\theta, \quad y = r\sin\theta \]

Calculus with Parametric and Polar Equations

For parametric equations \( x = f(t) \), \( y = g(t) \), the derivative is:

\[ \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} \]

For polar equations \( r = f(\theta) \), the derivative is:

\[ \frac{dy}{dx} = \frac{\frac{dr}{d\theta} \sin\theta + r \cos\theta}{\frac{dr}{d\theta} \cos\theta - r \sin\theta} \]

Examples

Example 1: Convert \( r = 2\cos\theta \) to Cartesian form.

\[ x = 2\cos\theta \cos\theta = 2\cos^2\theta \] \[ y = 2\cos\theta \sin\theta \]

Using \( \cos^2\theta = \frac{1 + \cos 2\theta}{2} \), we get:

\[ x^2 + y^2 = 2x \]

Exercises

  • Question 1: Convert the parametric equations \( x = 4t, y = t^2 \) into Cartesian form.
  • Question 2: Find the derivative \( \frac{dy}{dx} \) for the parametric equations \( x = t^2, y = t^3 \).
  • Question 3: Convert the polar equation \( r = 3\sin\theta \) to Cartesian form.

  • Answer 1: \( y = \frac{x^2}{16} \).
  • Answer 2: \( \frac{dy}{dx} = \frac{3t^2}{2t} = \frac{3t}{2} \).
  • Answer 3: \( x^2 + y^2 = 3y \).

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